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- ItemMultivariate regression models with measurement errors under generalized hyperbolic distributions(2025) Sandoval Moreno, Fanny Gabriela; Galea Rojas, Manuel Jesús; Pontificia Universidad Católica de Chile. Facultad de MatemáticasMeasurement error models (MEMs), also known as errors-in-variables models, are useful tools used to describe various phenomena across multiple disciplines. MEMs establish functional relationships among variables that are observed with random measurement errors. This thesis focuses on studying the multivariate linear regression model when covariates are measured with error (MRMMEs),extending the classical multivariate linear regression framework. It is important to note that calibration models (CCMs) are a sub-class of MRMMEs. Then, we also study an extension of CCMs.Although the normality assumption is useful in some cases, the multivariate normal distribution is not appropriate when the data comes from asymmetric or heavy-tailed distributions. In practice, multivariate data are often complex in the sense that they present high levels of asymmetry, kurtosis, and outliers. For these reasons, the study of probabilistic models and inferential aspects for continuous multivariate observations is of interest. Then, this thesis aims to carry out a statistical analysis of the multivariate regression model with measurement error by considering the multivariate normal and distributions belonging to the multivariate generalized hyperbolic (GH) family.One of the significant challenges we face was dealing with identifiability issues, particulary within the multivariate GH family of distributions. To address this, we focused on the symmetric case, which includes the symmetric normal inverse Gaussian (sym NIG) distribution and the symmetric hyperbolic (sym HYP) distribution. We used the Maximum Likelihood (ML) framework for estimation through the Expectation Maximization algorithm, hypothesis testing, model assessment, and influence diagnostics in both the multivariate regression model with measurement error under the normality assumption (N-MRMME) and the multivariate regression model with measurement error under multivariate symmetric generalized hyperbolic distributions (sym GH-MRMME). In both cases, we faced the challenge of complex expressions. However the suitable properties of the multivariate normal distribution and the GH distributions simplified the process.All proposed methodologies were illustrated with real data applications. Additionally, simulation studies based on the applications were conducted to assess the behavior of the estimation procedures, hypothesis tests, and influence diagnostics, providing support for its validity. The results showed accurate estimations that improved with larger sample sizes, as expected. Even in the presence of high levels of kurtosis, the symmetric GH models presented a better fit than the N-MRMME.Ultimately, this work contributes to the existing literature on measurement error models and generalized hyperbolic distributions.
- ItemEquilibrium states and asymptotic variance for geometric potentials(2025) Arévalo Hurtado, Nicolás; Iommi Echeverría, Godofredo; Pontificia Universidad Católica de Chile. Facultad de MatemáticasEn esta tesis abordamos tres problemas dentro del marco del formalismo termodinámico. En primer lugar, estudiamos el espectro de Lyapunov de las aplicaciones de Markov-Rényi-Lüroth en el intervalo, una familia de aplicaciones de intervalo de tipo Markov numerables que pueden presentar simultáneamente puntos fijos parabólicos y una discontinuidad en la presión topológica asociada al potencial geométrico. En segundo lugar, investigamos la existencia de estados de equilibrio para una familia de aplicaciones monótonas a trozos con convexidad promedio. Estas aplicaciones pueden tener puntos fijos parabólicos, particiones no Markovianas y un potencial geométrico que no es necesariamente Hölder continuo. Finalmente, analizamos la varianza asintótica para transformaciones abiertas, topológicamente transitivas y expansivas en espacios métricos compactos. Proporcionamos nuevas cotas para las diferencias en desigualdades de media potencia para potenciales Hölder continuos, expresadas en términos de la varianza asintótica.
- ItemStudy of attractors complexity in cellular automata and the topological structure of Z^d-subshifts(2025) Herrera Núñez, Alonso Eduardo; Rojas, Cristóbal; Sablik, Mathieu; Pontificia Universidad Católica de Chile. Facultad de MatemáticasThis work explores two notions of typicality in dynamical systems within the frameworkof symbolic dynamics. The first concerns typical systems from a topological viewpoint: thosethat are “large” in a topological sense, often referred to as generic systems. We focus onthe topological space S d of all subshifts AZd, where A is a finite subset of Z. For d = 1, R.Pavlov and S. Schmieding, [57], have shown that isolated subshifts are generic. To navigatethis question for d ≥ 2, we introduce the notion of maximal subsystem—a subsystem thatis inclusion-wise maximal—and use it to characterise isolated systems in Sd as follows: asubshift is isolated if and only if it is of finite type and it has a finite class of maximalsubsystems that contains every proper subsystem. This class is not generic as in the d = 1case, but any generic class must contain it, hence the interest in it. Later, we provideinsights into how the number of maximal subsystems a subshift has relates to its dynamicaland structural properties. Finally, we use some of the machinery developed to show that theCantor-Bendixon rank of Sd is infinite when d ≥ 2, which drastically differs from the casefor d = 1, where the Cantor-Bendixon rank is 1. The second notion changes focus to typically observable behaviours in the form of attractors. In the context of cellular automata, we investigate how complicated the generic and likely limit sets are, originally introduced by J. Milnor. To measure this, we rely on the well known arithmetical hierarchy and find that in general, the language of the likely limit set is a Σ3 set—for the generic limit set, the same upper bound was already known from [72]. Under the restriction of an automaton with equicontinuity points, we show that both attractors coincide and the complexity decreases to Σ1, with tight bounds. In the case the attractors are inclusion-wise minimal, we find an upper bound of Π2, matching known results for general systems, [62]. Finally, we prove the following realisation theorem: for any pair of chain-mixing Π2 subshifts Y ⊆ X, there exists a cellular automaton whose generic and likely limit sets are precisely X and Y , respectively. Altogether, this work offers new insights into the interplay between structure and observability in symbolic dynamical systems, highlighting how typical behaviours may vary dramatically across dimensions and under different dynamical constraints.
- ItemArithmetic of Drinfeld modules(2025) Alvarado Torres, Matías Nicolás; Pasten Vásquez, Héctor Hardy; Pontificia Universidad Católica de Chile. Facultad de MatemáticasDrinfeld modules, introduced by Vladimir Drinfeld in the 1970s, have become acornerstone in the arithmetic of global function fields. These objects serve as thefunction field analogues of elliptic curves and abelian varieties, but with a structure thatis uniquely adapted to the arithmetic of positive characteristic. Defined over rings offunctions rather than number fields, Drinfeld modules allow for the development of arich arithmetic theory that mirrors, and in many ways extends, the classical theory ofelliptic curves. Their moduli spaces, Galois representations, and associated L-functionshave all been studied extensively, revealing deep analogies with the number field caseand offering new phenomena unique to the function field setting. From an arithmeticstandpoint, Drinfeld modules provide explicit realizations of class field theory for globalfunction fields, particularly through the theory of Hayes modules and the use of shtukas.They give rise to Galois representations whose image and ramification behavior encodesignificant arithmetic information. Moreover, the theory of heights and canonical measuresassociated with Drinfeld modules has led to important results in Diophantine geometry,such as analogues of the Mordell-Weil theorem and the Bogomolov conjecture in positivecharacteristic. Beyond their arithmetic significance, Drinfeld modules also exhibit a richdynamical structure.
- ItemMathematical analysis and applications of neural networks, with applications to image reconstruction(2025) Molina Mejía, Juan José; Courdurier, Matías; Pontificia Universidad Católica de Chile. Facultad de MatemáticasThis thesis explores two fundamental aspects of neural networks: their frequency learning behavior and their application to quantitative Magnetic Resonance Imaging (MRI) reconstruction. The first part investigates the phenomenon of frequency bias, the empirical observation that neural networks tend to learn low-frequency components of a target function more rapidly than high-frequency ones. To provide a rigorous understanding of this behavior, we develop a theoretical framework based on Fourier analysis. Specifically, we derive a partial differential equation that governs the evolution of the error spectrum during training in the Neural Tangent Kernel regime, focusing on two-layer neural networks. Our analysis centers on Fourier Feature networks, a class of architectures where the first layer applies sine and cosine activations using pre-defined frequency distributions. We demonstrate that the network's initialization, particularly the initial density distribution of first-layer weights, plays a crucial role in shaping the frequency learning dynamics. This insight provides a principled way to control or even eliminate frequency bias during training. Theoretical predictions are validated through numerical experiments, which further illustrate the impact of initialization on the inductive biases of neural networks.The second part of the thesis applies neural network techniques to the reconstruction of quantitative MRI data. Quantitative MRI enables the estimation of tissue-specific parameters (e.g., T1, T2, and T2*) that are vital for clinical diagnosis and disease monitoring. However, these methods typically require long acquisition times, which are often mitigated through aggressive undersampling of k-space data. Undersampling, in turn, introduces reconstruction artifacts that must be addressed through regularization. To this end, we propose CConnect, a novel iterative reconstruction method that incorporates convolutional neural networks into the regularization term. CConnect connects multiple CNNs through a shared latent space, allowing the model to capture common structures across different image contrasts. This design enables the effective suppression of aliasing artifacts and improves image quality, even in highly undersampled scenarios. We evaluate CConnect on in-vivo brain T2*-weighted MRI data, demonstrating its superiority over classical low-rank and total variation methods, as well as standard deep learning baselines.